Chi-Square Simulation of the Cir Process and the Heston Model

Abstract

The transition probability of a Cox-Ingersoll-Ross process can be represented by a non-central chi-square density. First, we establish a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second, we show that Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance-rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third, we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley-Springer-Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.

Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

“…However, the issue of deriving a property-preserving numerical method when applying ABC to SDEs is usually seen as not so relevant, and it is usually recommended to use the Euler-Maruyama scheme or one of the higher order approximation methods described in Kloeden and Platen (1992); see, e.g., Picchini (2014); Picchini and Forman (2016); Picchini and Samson (2018); Sun et al (2015). In general, these standard methods do not preserve the underlying structural properties of the model; see, e.g., Ableidinger et al (2017); Malham and Wiese (2013); Moro and Schurz (2007); Strømmen Melbø and Higham (2004).…”

The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model

“…and the exponential is then incorporated in the payoff, i.e., g is replaced by f : R → R with f (x) = g(exp(x)). While exact simulation schemes and their refinements are known (see, e.g., Broadie and Kaya (2006); Glasserman and Kim (2011); Malham and Wiese (2013); Smith (2007)), discretization schemes as, e.g., Altmayer and Neuenkirch (2017); Andersen (2008); Kahl and Jäckel (2006); Lord et al (2009), are very popular for the Heston model. The latter discretization schemes can be easily extended to the multi-dimensional case and avoid computational bottlenecks of the exact schemes.…”

“…(i) exact simulations schemes which are based on the transition density of the CIR process and the characteristic function of the log-asset price process, see e.g. [12,28,17,24],…”

Estimation of ergodic square-root diffusion under high-frequency sampling

“…In this section, we compare simulation results of the initial estimator (2.5), (2.6), (2.7) with the one step estimators based on the Newton-Rhapson (2.17) and the scoring method (2.18). We use exact CIR simulator for (X tj ) j=1,...,n through non-central chi-squares [13]. The 1.000 simulated estimators are performed for n = 5.000, 10.000, 20.000 and T = T n = 500, 1.000, 2.000.…”